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- W4293443230 abstract "There are several equivalent ways to define continuous harmonic functions $H(K)$ on a compact set $K$ in $mathbb R^n$. One may let $H(K)$ be the unform closures of all functions in $C(K)$ which are restrictions of harmonic functions on a neighborhood of $K$, or take $H(K)$ as the subspace of $C(K)$ consisting of functions which are finely harmonic on the fine interior of $K$. In cite{DG74} it was shown that these definitions are equivalent. Using a localization result of cite{BH78} one sees that a function $hin H(K)$ if and only if it is continuous and finely harmonic on on every fine connected component of the fine interior of $K$. Such collection of sets are usually called {it restoring}. Another equivalent definition of $H(K)$ was introduced in cite{P97} using the notion of Jensen measures which leads another restoring collection of sets. The main goal of this paper is to reconcile the results in cite{DG74} and cite{P97}. To study these spaces, two notions of Green functions have previously been introduced. One by cite{P97} as the limit of Green functions on domains $D_j$ where the domains $D_j$ are decreasing to $K$, and alternatively following cite{F72, F75} one has the fine Green function on the fine interior of $K$. Our Theorem ref{T:green_equiv} shows that these are equivalent notions. In Section ref{S:Jensen} a careful study of the set of Jensen measures on $K$, leads to an interesting extension result (Corollary ref{C:extend}) for superharmonic functions. This has a number of applications. In particular we show that the two restoring coverings are the same. We are also able to extend some results of cite{GL83} and cite{P97} to higher dimensions." @default.
- W4293443230 created "2022-08-29" @default.
- W4293443230 creator A5046355862 @default.
- W4293443230 date "2010-12-17" @default.
- W4293443230 modified "2023-09-27" @default.
- W4293443230 title "Green functions, the fine topology and restoring coverings" @default.
- W4293443230 doi "https://doi.org/10.48550/arxiv.1012.3977" @default.
- W4293443230 hasPublicationYear "2010" @default.
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